Characterization of the limit load in the case of an unbounded elastic convex
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, p. 637-648

In this work we consider a solid body $\Omega \subset {ℝ}^{3}$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995) 391-419]. Then assuming that the convex of elasticity at the point x of $\Omega$, denoted by K(x), is written in the form of ${\text{K}}^{D}\left(x\right)+ℝ\text{I}$, I is the identity of ${{ℝ}^{9}}_{sym}$, and the deviatoric component ${\text{K}}^{D}$ is bounded regardless of x $\in \Omega$, we show under the condition “Rot f $\ne 0$ or g is not colinear to the normal on a part of the boundary of $\Omega$”, that the Limit Load $\overline{\lambda }$ searched is equal to the inverse of the infimum of the gauge of the Elastic convex translated by stress field equilibrating the unitary load corresponding to $\lambda =1$; moreover we show that this infimum is reached in a suitable function space.

DOI : https://doi.org/10.1051/m2an:2005028
Classification:  74xx
@article{M2AN_2005__39_4_637_0,
title = {Characterization of the limit load in the case of an unbounded elastic convex},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
pages = {637-648},
doi = {10.1051/m2an:2005028},
zbl = {pre02213933},
mrnumber = {2165673},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_4_637_0}
}

Elyacoubi, Adnene; Hadhri, Taieb. Characterization of the limit load in the case of an unbounded elastic convex. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, pp. 637-648. doi : 10.1051/m2an:2005028. http://www.numdam.org/item/M2AN_2005__39_4_637_0/

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] H. Brezis, Analyse Fonctionnelle. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[3] P.G. Ciarlet, Lectures on the three-dimensional elasticity. Tata Institute of Fundamental Research, Bombay (1983). | MR 730027 | Zbl 0542.73046

[4] H. El-Fekih and T. Hadhri, Calcul des charges limites d'une structure élastoplastique en contraintes planes. RAIRO: Modél. Math. Anal. Numér. 29 (1995) 391-419. | Numdam | Zbl 0831.73016

[5] R. Temam, Mathematical Problems in Plasticity. Bordas, Paris (1985).